College Math Teaching

March 7, 2012

Limits in a first course in multi-variable calculus

Filed under: analysis, calculus, pedagogy — collegemathteaching @ 1:47 am

Question: what is lim_{(x,y) \rightarrow (1,1)} \frac{x^2 - y^2}{x - y} ? The “obvious” answer is to rewrite \frac{x^2 - y^2}{x - y}  = x + y and then just substitute. The mathematical answer is to note that we do get a limit as we approach the point (1,1) from any point in \Delta \cap D where \Delta is a deleted disk and D is the domain of the function, which of course, does not include the line y = x . However, most calculus books just speak of the deleted disk without mentioning the domain; the interesting thing here is that the point (1,1) is NOT an isolated (if inessential) singularity.

This is a case in which being too precise might hinder understanding, but not being precise enough can lead to confusion.

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